One of the best hikes in all of Texas is the Guadalupe Peak trail in Guadalupe Mountains National Park. Not only can you see miles and miles of the Chihuahuan Desert floor below, but at the top of the trail, you really ARE at the highest point in Texas. (8,749 ft!)
Here's a link to the park's page on the various hiking trails, including Guadalupe Peak.
For more regional hiking information, visit: www.texasmountaintrail.com/hike